The original Pendant Appearances Theorem [1]}, [2]}, see also [3]}, [4]}, [5]}
says essentially that, if the set \(\mathcal {A}\)  of graphs has a growth constant, and the vertex-rooted connected graph \(H^{\bullet }\)  is attachable to \(\mathcal {A}\) , then there exists \(\beta >0\)  such that for \(R_n \in _u\mathcal {A}\)
\( \mathbb {P}( \rm pend(R_n,H^{\bullet }) \le \beta \, n) = e^{-\Omega (n)}.\) 
